naginterfaces.library.correg.linregm_​constrain

naginterfaces.library.correg.linregm_constrain(p, c, b, rss, idf)

linregm_constrain calculates the estimates of the parameters of a general linear regression model for given constraints from the singular value decomposition results.

For full information please refer to the NAG Library document for g02dk

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02dkf.html

Parameters

pfloat, array-like, shape $(\text{ip}\times \text{ip}+2\times \text{ip})$

As returned by linregm_fit() and linregm_update().

cfloat, array-like, shape $(\text{ip},\text{iconst})$

The $\text{iconst}$ constraints stored by column, i.e., the $i$th constraint is stored in the $i$th column of $c$.

bfloat, array-like, shape $\left(\text{ip}\right)$

The parameter estimates computed by using the singular value decomposition, ${\hat{\beta }}_{\text{svd}}$.

rssfloat

The residual sum of squares as returned by linregm_fit() or linregm_update().

idfint

The degrees of freedom associated with the residual sum of squares as returned by linregm_fit() or linregm_update().

Returns

bfloat, ndarray, shape $\left(\text{ip}\right)$

The parameter estimates of the parameters with the constraints imposed, ${\hat{\beta }}_c$.

sefloat, ndarray, shape $\left(\text{ip}\right)$

The standard error of the parameter estimates in $b$.

covfloat, ndarray, shape $(\text{ip}\times (\text{ip}+1)/2)$

The upper triangular part of the variance-covariance matrix of the $\text{ip}$ parameter estimates given in $b$. They are stored packed by column, i.e., the covariance between the parameter estimate given in $b[i-1]$ and the parameter estimate given in $b[j-1]$, $j\ge i$, is stored in $cov[(j\times (j-1)/2+i)-1]$.

Raises

NagValueError

(errno $1$)

On entry, $idf=⟨value⟩$.

Constraint: $idf>0$.

(errno $1$)

On entry, $\text{iconst}=⟨value⟩$.

Constraint: $\text{iconst}>0$.

(errno $1$)

On entry, $rss=⟨value⟩$.

Constraint: $rss>0.0$.

(errno $1$)

On entry, $\text{iconst}=⟨value⟩$ and $\text{ip}=⟨value⟩$.

Constraint: $\text{iconst}<\text{ip}$.

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

Constraint: $\text{ip}\ge 1$.

(errno $2$)

$c$ does not give a model of full rank.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

linregm_constrain computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to linregm_fit() or linregm_update().

In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates, ${\hat{\beta }}_{\text{svd}}$, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix $P^{\ast }$:

$$\begin{array}{c}{P}^{\ast }=\left(\begin{array}{c}{D}^{-1}{P}_1^T\\ P_0^T\end{array}\right)\text{,}\end{array}$$

as described by linregm_fit() and linregm_update().

Alternative solutions can be formed by imposing constraints on the parameters. If there are $p$ parameters and the rank of the model is $k$, then $n_c=p-k$ constraints will have to be imposed to obtain a unique solution.

Let $C$ be a $p\times n_c$ matrix of constraints, such that

$$C^T\beta =0$$

then the new parameter estimates ${\hat{\beta }}_c$ are given by

$$\begin{array}{c}\begin{array}{cc}{\hat{\beta }}_c& =A{\hat{\beta }}_{svd}\text{;}\\ & \\ & =(I-P_0{\left(C^T{P}_0\right)}^{-1}){\hat{\beta }}_{svd}\text{,}\end{array}\end{array}$$

where $I$ is the identity matrix, and the variance-covariance matrix is given by

$$A{P}_1{D}^{-2}{P}_1^T{A}^T\text{,}$$

provided ${\left(C^T{P}_0\right)}^{-1}$ exists.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

Hammarling, S, 1985, The singular value decomposition in multivariate statistics, SIGNUM Newsl. (20(3)), 2–25

Searle, S R, 1971, Linear Models, Wiley